Number Glyphs

The terms of the sequence are laid out in a grid, left-to-right and top to bottom. For each term, a glyph is generated: an image that depends on the term, and has distinctive visual features. The glyph generating algorithm may depend on all the terms shown on screen, but repeated terms in the sequence will give repeated glyphs.

The default glyph generation algorithm is as follows. Each glyph has a colour that reflects the prime factorization of the number, obtained by blending colours assigned to all the primes appearing as divisors in the terms of the sequence which appear on the screen. The term is drawn as a disk, whose brightness varies according to a given function from outer rim to center. The function grows faster for larger terms, and incorporates a modulus function so that one observes 'growth rings;' that is, tighter growth rings indicate a larger integer. Growth rings that are drawn more frequently than the pixel distance will be suffer from a sort of aliasing effect, appearing as if they are less frequent.

Bigint errors

Because math.js does not handle bigints, this visualizer will produce errors when any of the following occur:

• terms do not fit in the javascript Number type
• the growth function evaluated at a term does not fit in the Number type
• as with any visualizer, if the visualizer is used with a Sequence From Formula which produces invalid or incorrect output because of overflow

The latter two types of errors occur inside math.js and may produce unpredictable results. To handle the first type of error, currently this visualizer will set all terms whose absolute value exceeds $2^{53}-1$ to be 0.

Parameters

Number of Terms

The number of terms to display onscreen. The sizes of the discs will be sized so that there are $N^2$ disc positions, where $N^2$ is the smallest square exceeding the number of terms (so that the terms mostly fill the screen). Choose a perfect square number of terms to fill the square. If the sequence does not have that many terms, the visualizer will only attempt to show the available terms.

Customize Glyphs

This is a boolean which, if selected, will reveal further customization options for the glyph generation function.

Growth Function

This is a function in two variables, n and x. Most standard math notations are accepted (+, -, *, / , ^, log, sin, cos etc.) The variable n represents the term of which this disc is a representation. The variable x takes the value 0 at the outer rim of the disk, increasing once per pixel until the center. The absolute value of this function determines the brightness of the disk at that radius. A value of 0 is black and higher values are brighter.

The default value is log(max(abs(n),2)^x) % 25.

Brightness Adjustment

This is a brightness adjustment. It acts as a brightness cap or cutoff: brightness values will range from zero to this value. If set high (higher than most values of the function) it will produce dull dark glyphs. If set low, brightnesses above the cutoff will be rounded down to the cutoff, resulting in bright `flat' glyphs with less brightness variation.

In general, results are nice when the brightness adjustment is set to the maximum attained by the Growth Function.

The default value is 25.

Examples

Click on any image to expand it.

The Positive Integers

First, the non-negative integers, with the default settings. Next, with growth function taken
modulo 1 (instead of the default 25) and Brightness Adjustment set to 1. The second example shows some interesting effects because the rings occur more rapidly than once per pixel.

The semi-primes

This image shows the semi-primes (A001358). In this example, 121 elements are shown.

Another growth function

This shows the integers under the growth function $25(1-\cos(nx))$ modulo 25.

Credit

The original version of this visualizer was created by Jennifer Leong, as part of the Experimental Mathematics Lab at CU Boulder.